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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexStephan Spahn added stuff to _[[plus construction on presheaves]]_

   Stephan Spahn added stuff to plus construction on presheaves

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Stephan! you [ask](http://nforum.mathforge.org/discussion/3761/compact-objects-in-the-topos-over-all-manifolds/?Focus=30968#Comment_30968) > To address the question of compact objects in Sh ∞(SmoothMfd) there should be an (∞,1)-plus construction, too. Is in this case where the (∞,1)-site is just a 1-site somehow clear how this works? For $n$-truncated objects it is in principle clear: one has to apply the plus-construction (n+2)-times in a row! See for instance section 6.5.3 of [[Higher Topos Theory|HTT]].

   Thanks, Stephan!

   you ask

   To address the question of compact objects in Sh ∞(SmoothMfd) there should be an (∞,1)-plus construction, too. Is in this case where the (∞,1)-site is just a 1-site somehow clear how this works?

   For $n$-truncated objects it is in principle clear: one has to apply the plus-construction (n+2)-times in a row!

   See for instance section 6.5.3 of HTT.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a brief remark on this in the _[Idea section](http://ncatlab.org/nlab/show/plus+construction+on+presheaves#Idea)_. But am out of time now.

   I have added a brief remark on this in the Idea section. But am out of time now.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeApr 30th 2012
   • (edited Apr 30th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI have added the reference to the famous Heller-Rowe article where the plus construction in the case of abelian sheaves is studied. In the abelian context one usually says Heller-Rowe functor. Interestingly, the theory of Q-categories of Rosenberg has been written out in 1988 to exhibit the common and nontrivial generalization of the functor $H_{\mathcal{F}}$ in the theory of Gabriel localization and Heller/Rowe functor, as the instances of the same construction. Gabriel localization is also $H^2_{\mathcal{F}}$, just like sheafification. Here $\mathcal{F}$ is a Gabriel filter.

   I have added the reference to the famous Heller-Rowe article where the plus construction in the case of abelian sheaves is studied. In the abelian context one usually says Heller-Rowe functor. Interestingly, the theory of Q-categories of Rosenberg has been written out in 1988 to exhibit the common and nontrivial generalization of the functor $H_{ℱ}$ in the theory of Gabriel localization and Heller/Rowe functor, as the instances of the same construction. Gabriel localization is also $H_{ℱ}^2$, just like sheafification. Here $ℱ$ is a Gabriel filter.

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexI always thought that the fact that we needed to do $(-)^+$ twice had something to do with the fact that the equaliser diagram has two stages, but I never did find a good technical explanation of this point. What is clear, though, is that doing it once is a generalisation of computing $\check{H}^0$. I imagine these two facts are related via $n$-categories...

   I always thought that the fact that we needed to do $(-{)}^{+}$ twice had something to do with the fact that the equaliser diagram has two stages, but I never did find a good technical explanation of this point. What is clear, though, is that doing it once is a generalisation of computing ${\check{H}}^0$. I imagine these two facts are related via $n$-categories…